How Special Relativity Works

The Lorentz Transforms allow us to calculate the length contraction. How much contraction occurs is dependent on how fast an object is traveling with respect to the observer. Just to put some numbers to this, assume that a 12-inch football flies past you and it is moving at a rate of 60% the speed of light. You would measure the football to be 9.6 inches long. So at 60% the speed of light, you measure the football to be 80% of its original length (original 12 inch measurement was made at rest with respect to you). Keep in mind that all measurements are in the direction of the motion - The diameter of the ball is not changed by the ball's forward motion. Here are two points to keep in mind:

if you ran beside the football at the same speed, 60% the speed of light, you would always measure the length to be 12 inches. This is no different than you standing still and measuring the football while holding it.

if a lady running with the football measured a ruler that you are holding, she would measure you and your ruler to be length contracted as well. Remember, she has equal right to view you as being in motion with respect to her.

The Effect of Motion on Time

I mentioned that time also changes with different frames of reference (motion). This is known as "time dilation". Time actually slows with motion but it only becomes apparent at speeds close to the speed of light. Similar to length contraction, if the speed reaches that of light, time slows to a stop. Again, only an observer that is not in motion with the time that is being measured would notice. Like the tape measure in length contraction, a clock in motion would also be affected so it would never be able to detect that time was slowing down (remember the pendulum). Since our everyday motion does not approach anything remotely close to the speed of light, the dilation is completely unnoticed by us, but it is there.